You need passion to succeed. As long as you are thrilled by web programming, I don't think your age should be a problem. The only problem you might face, is tha

You need passion to succeed. As long as you are thrilled by web programming, I don't think your age should be a problem. The only problem you might face, is that you'll have to struggle a bit in the beginning. That's true for everybody, but it's easier for a teenager to live on an underpaid wage, than for someone who has a mortgage and kids to feed (Assuming you do).